2
$\begingroup$

Prove the following using the definition of a limit: $$\lim_{x \rightarrow -\infty} (1 + x^3) = -\infty.$$ I know we have to show that for all $x < N$, there must be $f(x) < M$, but I'm not quite sure what to do from here.

Any help would be greatly appreciated,

Thank you.

$\endgroup$
2
  • $\begingroup$ Could you, perhaps, show us what you've attempted? $\endgroup$ – Robin Goodfellow Oct 8 '14 at 16:25
  • $\begingroup$ If $x < -100$, what can you say about $1 + x^3$? $\endgroup$ – Pedro M. Oct 8 '14 at 16:28
1
$\begingroup$

Let $M < 0$ be given, choose N such that $N < \sqrt[3]{M-1}$. For $x < N$, we have: $1 + x^3 < 1 + (M - 1) = M$. This proves the result.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.